To calculate the standard deviation of the bus arrival times, we first need to determine the data points from the table and then compute the standard deviation using the following formula:
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Data Points: We can extract the data from the table for each arrival time along with their frequencies.
- -1 minutes: 1 day
- 0 minutes: 2 days
- 1 minute: 4 days
- 2 minutes: 2 days
- 3 minutes: 1 day
Based on this, we can list the arrival times:
- -1 (occurs 1 time)
- 0 (occurs 2 times)
- 1 (occurs 4 times)
- 2 (occurs 2 times)
- 3 (occurs 1 time)
This results in the numbers:
- -1, 0, 0, 1, 1, 1, 1, 2, 2, 3
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Calculate the Mean: \[ \text{Mean} (\mu) = \frac{\text{Sum of all data points}}{\text{Total number of data points}} = \frac{-1 + 0 + 0 + 1 + 1 + 1 + 1 + 2 + 2 + 3}{10} = \frac{10}{10} = 1 \]
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Calculate the Variance: \[ \text{Variance} (\sigma^2) = \frac{\sum (x - \mu)^2}{n} \] Where \( n \) is the total number of data points and \( x \) is each data point. \[ (-1 - 1)^2 = 4 \] \[ (0 - 1)^2 = 1 \] \[ (0 - 1)^2 = 1 \] \[ (1 - 1)^2 = 0 \] \[ (1 - 1)^2 = 0 \] \[ (1 - 1)^2 = 0 \] \[ (1 - 1)^2 = 0 \] \[ (2 - 1)^2 = 1 \] \[ (2 - 1)^2 = 1 \] \[ (3 - 1)^2 = 4 \] Now let's sum these values: \[ 4 + 1 + 1 + 0 + 0 + 0 + 0 + 1 + 1 + 4 = 12 \]
The total number of data points \( n = 10 \): \[ \text{Variance} = \frac{12}{10} = 1.2 \]
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Calculate the Standard Deviation: \[ \sigma = \sqrt{\sigma^2} = \sqrt{1.2} \approx 1.0954 \]
Rounding to the nearest tenth, we have: \[ \boxed{1.1} \]
Thus, the standard deviation is 1.1.